Ryan, I have a very similar issue on my 300SE that's been driving me nuts for over 10 years. Mine will start, then chug for about 5 seconds, and suddenly clear right up. So, it's not as bad as yours. My pressures hold well, just like yours, after shut-off.
Over the years, I've checked everything, and just can't find it.

From your pressure readings, it appears your system pressure may be too high. For my 6-cylinder, the system pressure is 5.5 Bar, and the pressure differential is 0.4 bar. It may be different for the V8, but you can do a search here to verify those numbers. The pressure controller would be the pressure regulator, but it doesn't seem likely one would read higher than spec as it got older???

You might also try a few of the things I've done (but found nothing wrong).

Disconnect the electrical connector on the cold start valve, or even unhook the line at the fuel distributor and plug it with an appropriate short bolt. This will eliminate the cold-start valve as either leaking or firing when it's not supposed to.

Check the mixture % duty cycle. There are several good tutorials herein.
basically get a cheap Sears meter with a duty cycle readout, and plug it across terminals 2&3 on the round diag connector. Should read close to 50% when warm and idling.

Make a jumper wire setup for the EHA plug, so you can patch in a current meter, and measure the EHA current. EHA should read about +-2 to 3 milliohms when idling warm. There are some good discriptions on doing this in the archives here also.

Check the engine vacuum when warm and idling. Should pull 15-20 inches.

None of these tests, and everything else I could think of in the last 10-15 years, have fixed my problem, but you might be luckier than me.
My engine runs absolutely perfect except for this, and gets great mileage.
All tests indicate spot-on, and everything seems to work just as it should, but it still chugs after a heat soak.

DG